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McKay matrices for finite-dimensional Hopf algebras

Part of: Hopf algebras, quantum groups and related topics Grothendieck groups and $K_0$

Published online by Cambridge University Press:  08 February 2021

Georgia Benkart ,
Rekha Biswal ,
Ellen Kirkman ,
Van C. Nguyen  and
Jieru Zhu
Georgia Benkart*
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Madison, WI53706, USA
Rekha Biswal
Affiliation:
Max Planck Institute for Mathematics, 53111Bonn, Germany e-mail: rekhabiswal27@gmail.com
Ellen Kirkman
Affiliation:
Department of Mathematics and Statistics, Wake Forest University, Winston-Salem, NC27109, USA e-mail: kirkman@wfu.edu
Van C. Nguyen
Affiliation:
Department of Mathematics, United States Naval Academy, Annapolis, MD21402, USA e-mail: vnguyen@usna.edu
Jieru Zhu
Affiliation:
Department of Mathematics, University at Buffalo, Buffalo, NY14260-2900, USA e-mail: jieruzhu699@gmail.com
*
e-mail: benkart@math.wisc.edu
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Abstract

For a finite-dimensional Hopf algebra $\mathsf {A}$ , the McKay matrix $\mathsf {M}_{\mathsf {V}}$ of an $\mathsf {A}$ -module $\mathsf {V}$ encodes the relations for tensoring the simple $\mathsf {A}$ -modules with $\mathsf {V}$ . We prove results about the eigenvalues and the right and left (generalized) eigenvectors of $\mathsf {M}_{\mathsf {V}}$ by relating them to characters. We show how the projective McKay matrix $\mathsf {Q}_{\mathsf {V}}$ obtained by tensoring the projective indecomposable modules of $\mathsf {A}$ with $\mathsf {V}$ is related to the McKay matrix of the dual module of $\mathsf {V}$ . We illustrate these results for the Drinfeld double $\mathsf {D}_n$ of the Taft algebra by deriving expressions for the eigenvalues and eigenvectors of $\mathsf {M}_{\mathsf {V}}$ and $\mathsf {Q}_{\mathsf {V}}$ in terms of several kinds of Chebyshev polynomials. For the matrix $\mathsf {N}_{\mathsf {V}}$ that encodes the fusion rules for tensoring $\mathsf {V}$ with a basis of projective indecomposable $\mathsf {D}_n$ -modules for the image of the Cartan map, we show that the eigenvalues and eigenvectors also have such Chebyshev expressions.

Keywords

McKay matrixDrinfeld doublecharacterChebyshev polynomial

MSC classification

Primary: 16T05: Hopf algebras and their applications
Secondary: 19A49: $K_0$ of other rings 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 3 , June 2022 , pp. 686 - 731
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Copyright
© Canadian Mathematical Society 2021

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